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In [[mathematics]], a '''closure operator''' on a [[Set (mathematics)|set]] ''S'' is a [[Function (mathematics)|function]] <math>\operatorname{cl}: \mathcal{P}(S)\rightarrow \mathcal{P}(S)</math> from the [[power set]] of ''S'' to itself that satisfies the following conditions for all sets <math>X,Y\subseteq S</math>
:{| border="0"
|-
| <math>X \subseteq \operatorname{cl}(X)</math>
| {{nb5}}(cl is ''extensive''),
|-
| <math>X\subseteq Y \Rightarrow \operatorname{cl}(X) \subseteq \operatorname{cl}(Y)</math>
| {{nb5}}(cl is ''increasing''),
|-
| <math> \operatorname{cl}(\operatorname{cl}(X))=\operatorname{cl}(X)</math>
| {{nb5}}(cl is ''idempotent'').
|}

Closure operators are determined by their '''closed sets''', i.e., by the sets of the form cl(''X''), since the '''closure''' cl(''X'') of a set ''X'' is the smallest closed set containing ''X''. Such families of "closed sets" are sometimes called '''closure systems''' or "'''Moore families'''".<ref>{{Cite journal |last=Diatta |first=Jean |date=2009-11-14 |title=On critical sets of a finite Moore family |url=https://doi.org/10.1007/s11634-009-0053-8 |journal=Advances in Data Analysis and Classification |language=en |volume=3 |issue=3 |pages=291–304 |doi=10.1007/s11634-009-0053-8 |s2cid=26138007 |issn=1862-5355}}</ref> A set together with a closure operator on it is sometimes called a '''closure space'''. Closure operators are also called "'''hull operators'''", which prevents confusion with the "closure operators" studied in [[point-set topology|topology]]. 

== History ==
[[E. H. Moore]] studied closure operators in his 1910 ''Introduction to a form of general analysis'', whereas the concept of the closure of a subset originated in the work of [[Frigyes Riesz]] in connection with topological spaces.{{sfn|Blyth|p=11}} Though not formalized at the time, the idea of closure originated in the late 19th century with notable contributions by [[Ernst Schröder (mathematician)|Ernst Schröder]], [[Richard Dedekind]] and [[Georg Cantor]].<ref>[[Marcel Erné]], ''Closure'', in Frédéric Mynard, Elliott Pearl 
(Editors), ''Beyond Topology'', Contemporary mathematics vol. 486, American Mathematical Society, 2009.</ref>

== Examples ==
[[File:Simple concave polygon Convex Hull.svg|thumb|Convex hull (red) of a [[polygon]] (yellow)]]
The usual set [[Closure (topology)|closure]] from [[topology]] is a closure operator. Other examples include the [[linear span]] of a subset of a [[vector space]], the [[convex hull]] or [[affine hull]] of a subset of a vector space or the [[lower semicontinuous]] hull <math>\overline{f}</math> of a function <math>f \colon E \to \mathbb{R} \cup \{ \pm \infty \}</math>, where <math>E</math> is e.g. a [[normed space]], defined implicitly <math>\operatorname{epi}(\overline{f}) = \overline{\operatorname{epi}(f)}</math>, where <math>\operatorname{epi}(f)</math> is the [[Epigraph (mathematics)|epigraph]] of a function <math>f</math>.

The [[relative interior]] <math>\operatorname{ri}</math> is not a closure operator: although it is idempotent, it is not increasing and if <math>C_1</math> is a cube in <math>\mathbb{R}^3</math> and <math>C_2</math> is one of its faces, then <math>C_2 \subset C_1</math>, but <math>\operatorname{ri}(C_1) \ne \emptyset \ne \operatorname{ri}(C_2)</math> and <math>\operatorname{ri}(C_1) \cap \operatorname{ri}(C_2) = \emptyset</math>, so it is not increasing.<ref>{{cite book |last1=Rockafellar |first1=Ralph Tyrell |title=Convex Analysis |date=1970 |publisher=Princeton University Press |isbn=9781400873173 |page=44 |doi=10.1515/9781400873173 |url=https://doi.org/10.1515/9781400873173}}</ref>

In topology, the closure operators are [[Kuratowski closure axioms|''topological'' closure operator]]s, which must satisfy

: <math>\operatorname{cl}(X_1 \cup\dots\cup X_n) = \operatorname{cl}(X_1)\cup\dots\cup \operatorname{cl}(X_n)</math>

for all <math>n\in\N</math> (Note that for <math>n=0</math> this gives <math>\operatorname{cl}(\varnothing)=\varnothing</math>).

In [[algebra]] and [[logic]], many closure operators are '''finitary closure operators''', i.e. they satisfy

: <math>\operatorname{cl}(X) = \bigcup\left\{\operatorname{cl}(Y) : Y\subseteq X \text{ and } Y \text{ finite} \right\}.</math>

In the theory of [[partially ordered set]]s, which are important in [[theoretical computer science]], closure operators have a more general definition that replaces <math>\subseteq</math> with <math>\leq</math>. (See {{section link||Closure operators on partially ordered sets}}.)

== Closure operators in topology ==
{{main|Kuratowski closure axioms}}
The [[topological closure]] of a subset ''X'' of a [[topological space]] consists of all points ''y'' of the space, such that every [[neighbourhood (mathematics)|neighbourhood]] of ''y'' contains a point of ''X''. The function that associates to every subset ''X'' its closure is a topological closure operator. Conversely, every topological closure operator on a set gives rise to a topological space whose closed sets are exactly the closed sets with respect to the closure operator.

== Closure operators in algebra ==
Finitary closure operators play a relatively prominent role in [[universal algebra]], and in this context they are traditionally called ''algebraic closure operators''. Every subset of an [[structure (mathematical logic)|algebra]] ''generates'' a [[substructure (mathematics)|subalgebra]]: the smallest subalgebra containing the set. This gives rise to a finitary closure operator.

Perhaps the best known example for this is the function that associates to every subset of a given [[vector space]] its [[linear span]]. Similarly, the function that associates to every subset of a given [[group (mathematics)|group]] the [[subgroup]] generated by it, and similarly for [[field (mathematics)|field]]s and all other types of [[algebraic structure]]s.

The linear span in a vector space and the similar algebraic closure in a field both satisfy the ''exchange property:'' If ''x'' is in the closure of the union of ''A'' and {''y''} but not in the closure of ''A'', then ''y'' is in the closure of the union of ''A'' and {''x''}. A finitary closure operator with this property is called a [[matroid]]. The [[dimension (vector space)|dimension]] of a vector space, or the [[transcendence degree]] of a field (over its [[prime field]]) is exactly the rank of the corresponding matroid.

The function that maps every subset of a given [[field (mathematics)|field]] to its [[algebraic closure]] is also a finitary closure operator, and in general it is different from the operator mentioned before. Finitary closure operators that generalize these two operators are studied in [[model theory]] as dcl (for ''definable closure'') and acl (for ''algebraic closure'').

The [[convex hull]] in ''n''-dimensional [[Euclidean space]] is another example of a finitary closure operator. It satisfies the ''anti-exchange property:'' If ''x'' is in the closure of the union of {''y''} and ''A'', but not in the union of {''y''} and closure of ''A'', then ''y'' is not in the closure of the union of {''x''} and ''A''. Finitary closure operators with this property give rise to [[antimatroid]]s.

As another example of a closure operator used in algebra, if some algebra has universe ''A'' and ''X'' is a set of pairs of ''A'', then the operator assigning to ''X'' the smallest [[congruence relation|congruence]] containing ''X'' is a finitary closure operator on ''A x A''.<ref>Clifford Bergman, ''Universal Algebra'', 2012, 
Section 2.4.</ref>

== Closure operators in logic ==
Suppose you have some [[mathematical logic|logical formalism]] that contains certain rules allowing you to derive new formulas from given ones. Consider the set ''F'' of all possible formulas, and let ''P'' be the [[power set]] of ''F'', ordered by ⊆. For a set ''X'' of formulas, let cl(''X'') be the set of all formulas that can be derived from ''X''. Then cl is a closure operator on ''P''. More precisely, we can obtain cl as follows. Call "continuous" an operator ''J'' such that, for every [[directed set|directed]] class ''T'',
	
:''J''(lim ''T'') = lim ''J''(''T'').	

This continuity condition is on the basis of a fixed point theorem for ''J''. Consider the one-step operator ''J'' of a monotone logic. This is the operator associating any set ''X'' of formulas with the set ''J''(''X'') of formulas that are either logical axioms or are obtained by an inference rule from formulas in ''X'' or are in ''X''. Then such an operator is continuous and we can define cl(''X'') as the least fixed point for ''J'' greater or equal to ''X''. In accordance with such a point of view, Tarski, Brown, Suszko and other authors proposed a general approach to logic based on closure operator theory. Also, such an idea is proposed in programming logic (see Lloyd 1987) and in [[fuzzy logic]] (see Gerla 2000).

=== Consequence operators ===
Around 1930, [[Alfred Tarski]] developed an abstract theory of logical deductions that models some properties of logical calculi. Mathematically, what he described is just a finitary closure operator on a set (the set of ''sentences'').  In [[abstract algebraic logic]], finitary closure operators are still studied under the name ''consequence operator'', which was coined by Tarski. The set ''S'' represents a set of sentences, a subset ''T'' of ''S'' a theory, and cl(''T'') is the set of all sentences that follow from the theory. Nowadays the term can refer to closure operators that need not be finitary; finitary closure operators are then sometimes called '''finite consequence operators'''.

== Closed sets ==
The closed sets with respect to a closure operator on ''S'' form a subset ''C'' of the power set '''''P'''''(''S''). Any intersection of sets in ''C'' is again in ''C''. In other words, ''C'' is a complete [[meet-semilattice|meet-subsemilattice]] of '''''P'''''(''S''). Conversely, if ''C'' ⊆ '''''P'''''(''S'') is closed under arbitrary intersections, then the function that associates to every subset ''X'' of ''S'' the smallest set ''Y'' ∈ ''C'' such that ''X'' ⊆ ''Y'' is a closure operator.

There is a simple and fast algorithm for generating all closed sets of a given closure operator.<ref>Ganter, Algorithm 1</ref>

A closure operator on a set is topological if and only if the set of closed sets is closed under finite unions, i.e., ''C'' is a meet-complete [[sublattice]] of '''''P'''''(''S''). Even for non-topological closure operators, ''C'' can be seen as having the structure of a lattice. (The join of two sets ''X'',''Y'' ⊆ '''''P'''''(''S'') being cl(''X'' <math>\cup</math> ''Y'').) But then ''C'' is not a sublattice of the lattice '''''P'''''(''S'').

Given a finitary closure operator on a set, the closures of finite sets are exactly the [[compact element]]s of the set ''C'' of closed sets. It follows that ''C'' is an [[algebraic poset]].
Since ''C'' is also a lattice, it is often referred to as an algebraic lattice in this context. Conversely, if ''C'' is an algebraic poset, then the closure operator is finitary.

=== Pseudo-closed sets ===
Each closure operator on a finite set ''S'' is uniquely determined by its images of its ''pseudo-closed'' sets.<ref>Ganter, Section 3.2</ref>
These are recursively defined: A set is '''pseudo-closed''' if it is not closed and contains the closure of each of its pseudo-closed proper subsets. Formally: ''P''&nbsp;&sube;&nbsp;''S'' is pseudo-closed if and only if
* ''P''&nbsp;&ne;&nbsp;cl(''P'') and 
* if ''Q''&nbsp;&sub;&nbsp;''P'' is pseudo-closed, then cl(''Q'')&nbsp;&sube;&nbsp;''P''.

== Closure operators on partially ordered sets ==
A [[partially ordered set]] (poset) is a set together with a ''partial order'' ≤, i.e. a [[binary relation]] that is reflexive ({{nowrap|''a'' ≤ ''a''}}), transitive ({{nowrap|''a'' ≤ ''b'' ≤ ''c''}} implies {{nowrap|''a'' ≤ ''c''}}) and [[Antisymmetric relation|antisymmetric]] ({{nowrap|''a'' ≤ ''b'' ≤ ''a''}} implies ''a''&nbsp;=&nbsp;''b''). Every [[power set]] '''P'''(''S'') together with inclusion ⊆ is a partially ordered set.

A function cl: ''P'' → ''P'' from a partial order ''P'' to itself is called a closure operator if it satisfies the following axioms for all elements ''x'', ''y'' in ''P''.
:{| border="0"
|-
| ''x'' ≤ cl(''x'')
| (cl is ''extensive'')
|-
| ''x'' ≤ ''y'' implies cl(''x'') ≤ cl(''y'')&nbsp;&nbsp;
| (cl is [[increasing]])
|-
| cl(cl(''x'')) = cl(''x'')
| (cl is [[idempotent]])
|}

More succinct alternatives are available: the definition above is equivalent to the single axiom

:''x'' ≤ cl(''y'') if and only if cl(''x'') ≤ cl(''y'')

for all ''x'', ''y'' in ''P''.

Using the [[pointwise order]] on functions between posets, one may alternatively write the extensiveness property as id<sub>''P''</sub> ≤ cl, where id is the [[identity function]]. A self-map ''k'' that is increasing and idempotent, but satisfies the [[Duality (order theory)|dual]] of the extensiveness property, i.e. ''k'' ≤ id<sub>''P''</sub> is called a '''kernel operator''',<ref>Giertz, p. 26</ref> '''interior operator''',<ref>Erné, p. 2, uses closure (resp. interior) operation</ref> or '''dual closure'''.<ref>Blyth, p. 10</ref> As examples, if ''A'' is a subset of a set ''B'', then the self-map on the powerset of ''B'' given by ''μ<sub>A</sub>''(''X'') = ''A'' ∪ ''X'' is a closure operator, whereas ''λ<sub>A</sub>''(''X'') = ''A'' ∩ ''X'' is a kernel operator. The [[ceiling function]] from the [[real number]]s to the real numbers, which assigns to every real ''x'' the smallest [[integer]] not smaller than ''x'', is another example of a closure operator.

A [[fixpoint]] of the function cl, i.e. an element ''c'' of ''P'' that satisfies cl(''c'')&nbsp;=&nbsp;''c'', is called a '''closed element'''. A closure operator on a partially ordered set is determined by its closed elements. If ''c'' is a closed element, then ''x'' ≤ ''c'' and cl(''x'') ≤ ''c'' are equivalent conditions.

Every [[Galois connection]] (or [[residuated mapping]]) gives rise to a closure operator (as is explained in that article). In fact, ''every'' closure operator arises in this way from a suitable Galois connection.<ref>Blyth, p. 10</ref> The Galois connection is not uniquely determined by the closure operator. One Galois connection that gives rise to the closure operator cl can be described as follows: if ''A'' is the set of closed elements with respect to cl, then cl: ''P'' → ''A'' is the lower adjoint of a Galois connection between ''P'' and ''A'', with the upper adjoint being the embedding of ''A'' into ''P''. Furthermore, every lower adjoint of an embedding of some subset into ''P'' is a closure operator. "Closure operators are lower adjoints of embeddings." Note however that not every embedding has a lower adjoint.

Any partially ordered set ''P'' can be viewed as a [[category theory|category]], with a single morphism from ''x'' to ''y'' if and only if ''x'' ≤ ''y''. The closure operators on the partially ordered set ''P'' are then nothing but the [[monad (category theory)|monad]]s on the category ''P''. Equivalently, a closure operator can be viewed as an endofunctor on the category of partially ordered sets that has the additional ''idempotent'' and ''extensive'' properties.

If ''P'' is a [[complete lattice]], then a subset ''A'' of ''P'' is the set of closed elements for some closure operator on ''P'' if and only if ''A'' is a '''Moore family''' on ''P'', i.e. the largest element of ''P'' is in ''A'', and the [[infimum]] (meet) of any non-empty subset of ''A'' is again in ''A''. Any such set ''A'' is itself a complete lattice with the order inherited from ''P'' (but the [[supremum]] (join) operation might differ from that of ''P''). When ''P'' is the [[powerset]] [[Boolean algebra]] of a set ''X'', then a Moore family on ''P'' is called a '''closure system''' on ''X''.

The closure operators on ''P'' form themselves a complete lattice; the order on closure operators is defined by cl<sub>1</sub> ≤ cl<sub>2</sub> [[iff]] cl<sub>1</sub>(''x'') ≤ cl<sub>2</sub>(''x'') for all ''x'' in ''P''.

==See also==

* {{annotated link|Čech closure operator}}
* {{annotated link|Closure (topology)}}
* {{annotated link|Galois connection}}
* {{annotated link|Interior algebra}}
* {{annotated link|Interior (topology)}}
* {{annotated link|Kuratowski closure axioms}}
* {{annotated link|Preclosure operator}}

== Notes ==
{{reflist|3}}

== References ==
* [[Garrett Birkhoff]]. 1967 (1940). ''Lattice Theory, 3rd ed''. American Mathematical Society.
* Burris, Stanley N., and H.P. Sankappanavar (1981) [http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra]  Springer-Verlag. {{ISBN|3-540-90578-2}} ''Free online edition''.
* Brown, D.J. and Suszko, R. (1973) "Abstract Logics," [[Dissertationes Mathematicae]] 102- 9-42.
* Castellini, G. (2003) ''Categorical closure operators''. Boston MA: Birkhaeuser.
* Edelman, Paul H. (1980) ''Meet-distributive lattices and the anti-exchange closure,'' [[Algebra Universalis]] 10: 290-299.
* Ganter, Bernhard and Obiedkov, Sergei (2016) ''Conceptual Exploration''. Springer, {{ISBN|978-3-662-49290-1}}.
* Gerla, G. (2000) ''Fuzzy Logic: Mathematical Tools for Approximate Reasoning''. [[Kluwer Academic Publishers]].
* Lloyd, J.W. (1987) ''Foundations of Logic Programming''. [[Springer-Verlag]].
* [[Alfred Tarski|Tarski, Alfred]] (1983) "Fundamental concepts of the methodology of deductive sciences" in ''Logic, Semantics, Metamathematics''. Hackett (1956 ed., [[Oxford University Press]]).
* [[Alfred Tarski]] (1956) ''Logic, semantics and metamathematics''. [[Oxford University Press]].
* [[Morgan Ward|Ward, Morgan]] (1942) "The closure operators of a lattice," [[Annals of Mathematics]] 43: 191-96.
* G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott: ''Continuous Lattices and Domains'', Cambridge University Press, 2003
* {{wikicite|reference=T.S. Blyth, ''Lattices and Ordered Algebraic Structures'', Springer, 2005, {{ISBN|1-85233-905-5}}.|ref={{harvid|Blyth}}}}
* M. Erné, J. Koslowski, A. Melton, G. E. Strecker, ''A primer on Galois connections'', in: Proceedings of the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work, Annals of the New York Academy of Sciences, Vol. 704, 1993, pp.&nbsp;103–125. Available online in various file formats: [https://web.archive.org/web/20060108063506/http://www.iti.cs.tu-bs.de/TI-INFO/koslowj/RESEARCH/gal_bw.ps.gz PS.GZ] [http://www.math.ksu.edu/~strecker/primer.ps PS]

==External links==
*[[Stanford Encyclopedia of Philosophy]]: "[https://plato.stanford.edu/entries/logic-algebraic-propositional/ Algebraic Propositional Logic]"—by Ramon Jansana.

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